the Loewner differential equation

Suppose you want to understand a single complicated univalent map. A powerful trick is not to study it frozen but to GROW it: build a continuous movie of univalent maps, starting from the trivial identity and slowly deforming until you reach your target, and track how the map changes from frame to frame. The rule governing that frame-to-frame change is a differential equation. Loewner discovered the right such equation in 1923, and it became the master tool for the hardest coefficient problems.

The setup is a Loewner chain: a family f_t of univalent functions on the disk, indexed by a time t >= 0, with f_0(z) = z and the images steadily expanding as t grows (each earlier image sits inside each later one). For the slit version, where at each instant a tip is being pushed in from the boundary, the maps obey the radial Loewner differential equation. In the half-plane chordal form it reads, for the inverse maps, dg_t(z)/dt = 2 / (g_t(z) - U_t), where U_t is a real 'driving function' running along the boundary that encodes which boundary point is being absorbed at time t. Solving the ODE forward reconstructs the whole growing family from the single driving function; the geometry of the map is repackaged as the dynamics of one boundary point.

This is exactly the machinery behind the deep coefficient theorems: Loewner himself used it to prove |a_3| <= 3, and de Branges built his proof of the full Bieberbach conjecture on Loewner chains. In the twenty-first century the same equation, fed a Brownian-motion driving function, became Schramm-Loewner evolution (SLE), a cornerstone of probability and statistical physics. A caution: the driving-function correspondence is delicate — a continuous U_t produces a growing hull, but not every U_t yields a slit by a smooth curve, and reading geometry off U_t (or vice versa) is genuinely subtle, not a simple dictionary.